Optimal. Leaf size=45 \[ -\frac {\sqrt [4]{x^3+1}}{3 x^3}-\frac {1}{6} \tan ^{-1}\left (\sqrt [4]{x^3+1}\right )-\frac {1}{6} \tanh ^{-1}\left (\sqrt [4]{x^3+1}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {266, 47, 63, 212, 206, 203} \begin {gather*} -\frac {\sqrt [4]{x^3+1}}{3 x^3}-\frac {1}{6} \tan ^{-1}\left (\sqrt [4]{x^3+1}\right )-\frac {1}{6} \tanh ^{-1}\left (\sqrt [4]{x^3+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 203
Rule 206
Rule 212
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{1+x^3}}{x^4} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt [4]{1+x}}{x^2} \, dx,x,x^3\right )\\ &=-\frac {\sqrt [4]{1+x^3}}{3 x^3}+\frac {1}{12} \operatorname {Subst}\left (\int \frac {1}{x (1+x)^{3/4}} \, dx,x,x^3\right )\\ &=-\frac {\sqrt [4]{1+x^3}}{3 x^3}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\sqrt [4]{1+x^3}\right )\\ &=-\frac {\sqrt [4]{1+x^3}}{3 x^3}-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{1+x^3}\right )-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{1+x^3}\right )\\ &=-\frac {\sqrt [4]{1+x^3}}{3 x^3}-\frac {1}{6} \tan ^{-1}\left (\sqrt [4]{1+x^3}\right )-\frac {1}{6} \tanh ^{-1}\left (\sqrt [4]{1+x^3}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 26, normalized size = 0.58 \begin {gather*} \frac {4}{15} \left (x^3+1\right )^{5/4} \, _2F_1\left (\frac {5}{4},2;\frac {9}{4};x^3+1\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 45, normalized size = 1.00 \begin {gather*} -\frac {\sqrt [4]{1+x^3}}{3 x^3}-\frac {1}{6} \tan ^{-1}\left (\sqrt [4]{1+x^3}\right )-\frac {1}{6} \tanh ^{-1}\left (\sqrt [4]{1+x^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 57, normalized size = 1.27 \begin {gather*} -\frac {2 \, x^{3} \arctan \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}}\right ) + x^{3} \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}} + 1\right ) - x^{3} \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}} - 1\right ) + 4 \, {\left (x^{3} + 1\right )}^{\frac {1}{4}}}{12 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 48, normalized size = 1.07 \begin {gather*} -\frac {{\left (x^{3} + 1\right )}^{\frac {1}{4}}}{3 \, x^{3}} - \frac {1}{6} \, \arctan \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{12} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{12} \, \log \left ({\left | {\left (x^{3} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.59, size = 52, normalized size = 1.16
method | result | size |
meijerg | \(-\frac {\frac {4 \Gamma \left (\frac {3}{4}\right )}{x^{3}}-\left (-3 \ln \relax (2)+\frac {\pi }{2}-1+3 \ln \relax (x )\right ) \Gamma \left (\frac {3}{4}\right )+\frac {3 \hypergeom \left (\left [1, 1, \frac {7}{4}\right ], \left [2, 3\right ], -x^{3}\right ) \Gamma \left (\frac {3}{4}\right ) x^{3}}{8}}{12 \Gamma \left (\frac {3}{4}\right )}\) | \(52\) |
risch | \(-\frac {\left (x^{3}+1\right )^{\frac {1}{4}}}{3 x^{3}}+\frac {\left (-3 \ln \relax (2)+\frac {\pi }{2}+3 \ln \relax (x )\right ) \Gamma \left (\frac {3}{4}\right )-\frac {3 \hypergeom \left (\left [1, 1, \frac {7}{4}\right ], \left [2, 2\right ], -x^{3}\right ) \Gamma \left (\frac {3}{4}\right ) x^{3}}{4}}{12 \Gamma \left (\frac {3}{4}\right )}\) | \(56\) |
trager | \(-\frac {\left (x^{3}+1\right )^{\frac {1}{4}}}{3 x^{3}}-\frac {\ln \left (\frac {2 \left (x^{3}+1\right )^{\frac {3}{4}}+x^{3}+2 \sqrt {x^{3}+1}+2 \left (x^{3}+1\right )^{\frac {1}{4}}+2}{x^{3}}\right )}{12}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{3}+1}+2 \left (x^{3}+1\right )^{\frac {3}{4}}-2 \RootOf \left (\textit {\_Z}^{2}+1\right )-2 \left (x^{3}+1\right )^{\frac {1}{4}}}{x^{3}}\right )}{12}\) | \(119\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 47, normalized size = 1.04 \begin {gather*} -\frac {{\left (x^{3} + 1\right )}^{\frac {1}{4}}}{3 \, x^{3}} - \frac {1}{6} \, \arctan \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{12} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{12} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.54, size = 33, normalized size = 0.73 \begin {gather*} -\frac {\mathrm {atan}\left ({\left (x^3+1\right )}^{1/4}\right )}{6}-\frac {\mathrm {atanh}\left ({\left (x^3+1\right )}^{1/4}\right )}{6}-\frac {{\left (x^3+1\right )}^{1/4}}{3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.90, size = 34, normalized size = 0.76 \begin {gather*} - \frac {\Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{3}}} \right )}}{3 x^{\frac {9}{4}} \Gamma \left (\frac {7}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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